Alexander Holmes; Barbara Illowsky; Susan Dean

English Phrases Written Mathematically

When the English says:	Interpret this as:
X is at least 4.	X ≥ 4
The minimum of X is 4.	X ≥ 4
X is no less than 4.	X ≥ 4
X is greater than or equal to 4.	X ≥ 4
X is at most 4.	X ≤ 4
The maximum of X is 4.	X ≤ 4
X is no more than 4.	X ≤ 4
X is less than or equal to 4.	X ≤ 4
X does not exceed 4.	X ≤ 4
X is greater than 4.	X > 4
X is more than 4.	X > 4
X exceeds 4.	X > 4
X is less than 4.	X < 4
There are fewer X than 4.	X < 4
X is 4.	X = 4
X is equal to 4.	X = 4
X is the same as 4.	X = 4
X is not 4.	X ≠ 4
X is not equal to 4.	X ≠ 4
X is not the same as 4.	X ≠ 4
X is different than 4.	X ≠ 4

Table B1

Symbols and Their Meanings

Chapter (1st used)	Symbol	Spoken	Meaning
Sampling and Data	$\sqrt{\begin{matrix} \end{matrix}}$	The square root of	same
Sampling and Data	$π$	Pi	3.14159… (a specific number)
Descriptive Statistics	Q₁	Quartile one	the first quartile
Descriptive Statistics	Q₂	Quartile two	the second quartile
Descriptive Statistics	Q₃	Quartile three	the third quartile
Descriptive Statistics	IQR	interquartile range	Q₃ – Q₁ = IQR
Descriptive Statistics	$\bar{x}$	x-bar	sample mean
Descriptive Statistics	$μ$	mu	population mean
Descriptive Statistics	s	s	sample standard deviation
Descriptive Statistics	$s^{2}$	s squared	sample variance
Descriptive Statistics	$σ$	sigma	population standard deviation
Descriptive Statistics	$σ^{2}$	sigma squared	population variance
Descriptive Statistics	$Σ$	capital sigma	sum
Probability Topics	${}$	brackets	set notation
Probability Topics	$S$	S	sample space
Probability Topics	$A$	Event A	event A
Probability Topics	$P (A)$	probability of A	probability of A occurring
Probability Topics	$P (A \| B)$	probability of A given B	prob. of A occurring given B has occurred
Probability Topics	$P (A \cup B)$	prob. of A or B	prob. of A or B or both occurring
Probability Topics	$P (A \cap B)$	prob. of A and B	prob. of both A and B occurring (same time)
Probability Topics	A′	A-prime, complement of A	complement of A, not A
Probability Topics	P(A')	prob. of complement of A	same
Probability Topics	G₁	green on first pick	same
Probability Topics	P(G₁)	prob. of green on first pick	same
Discrete Random Variables	PDF	prob. density function	same
Discrete Random Variables	X	X	the random variable X
Discrete Random Variables	X ~	the distribution of X	same
Discrete Random Variables	$\geq$	greater than or equal to	same
Discrete Random Variables	$\leq$	less than or equal to	same
Discrete Random Variables	=	equal to	same
Discrete Random Variables	≠	not equal to	same
Continuous Random Variables	f(x)	f of x	function of x
Continuous Random Variables	pdf	prob. density function	same
Continuous Random Variables	U	uniform distribution	same
Continuous Random Variables	Exp	exponential distribution	same
Continuous Random Variables	f(x) =	f of x equals	same
Continuous Random Variables	m	m	decay rate (for exp. dist.)
The Normal Distribution	N	normal distribution	same
The Normal Distribution	z	z-score	same
The Normal Distribution	Z	standard normal dist.	same
The Central Limit Theorem	$\bar{X}$	X-bar	the random variable X-bar
The Central Limit Theorem	$μ_{\bar{x}}$	mean of X-bars	the average of X-bars
The Central Limit Theorem	$σ_{\bar{x}}$	standard deviation of X-bars	same
Confidence Intervals	CL	confidence level	same
Confidence Intervals	CI	confidence interval	same
Confidence Intervals	EBM	error bound for a mean	same
Confidence Intervals	EBP	error bound for a proportion	same
Confidence Intervals	t	Student's t-distribution	same
Confidence Intervals	df	degrees of freedom	same
Confidence Intervals	$t_{\frac{α}{2}}$	student t with α/2 area in right tail	same
Confidence Intervals	$p'$	p-prime	sample proportion of success
Confidence Intervals	$q'$	q-prime	sample proportion of failure
Hypothesis Testing	$H_{0}$	H-naught, H-sub 0	null hypothesis
Hypothesis Testing	$H_{a}$	H-a, H-sub a	alternate hypothesis
Hypothesis Testing	$H_{1}$	H-1, H-sub 1	alternate hypothesis
Hypothesis Testing	$α$	alpha	probability of Type I error
Hypothesis Testing	$β$	beta	probability of Type II error
Hypothesis Testing	$\bar{X 1} - \bar{X 2}$	X1-bar minus X2-bar	difference in sample means
Hypothesis Testing	$μ_{1} - μ_{2}$	mu-1 minus mu-2	difference in population means
Hypothesis Testing	${P^{'}}_{1} - {P^{'}}_{2}$	P1-prime minus P2-prime	difference in sample proportions
Hypothesis Testing	$p_{1} - p_{2}$	p1 minus p2	difference in population proportions
Chi-Square Distribution	$Χ^{2}$	Ky-square	Chi-square
Chi-Square Distribution	$O$	Observed	Observed frequency
Chi-Square Distribution	$E$	Expected	Expected frequency
Linear Regression and Correlation	y = a + bx	y equals a plus b-x	equation of a straight line
Linear Regression and Correlation	$\hat{y}$	y-hat	estimated value of y
Linear Regression and Correlation	$r$	sample correlation coefficient	same
Linear Regression and Correlation	$ε$	error term for a regression line	same
Linear Regression and Correlation	SSE	Sum of Squared Errors	same
F-Distribution and ANOVA	F	F-ratio	F-ratio

Table B2 Symbols and their Meanings

Formulas

Symbols you must know

Population

Sample

N

Size

n

μ

Mean

\overline{x}

σ^{2}

Variance

s^{2}

σ

Standard deviation

s

p

Proportion

p ′

Single data set formulae

Population

Sample

μ = E (x) = \frac{1}{N} \sum_{i = 1}^{N} (x_{i})

Arithmetic mean

\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} (x_{i})

Geometric mean

\tilde{x} = {(\prod_{i = 1}^{n} X_{i})}^{\frac{1}{n}}

Q_{3} = \frac{3 (n + 1)}{4}

,

Q_{1} = \frac{(n + 1)}{4}

Inter-quartile range

I Q R = Q_{3} - Q_{1}

Q_{3} = \frac{3 (n + 1)}{4}

,

Q_{1} = \frac{(n + 1)}{4}

σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} {(x_{i} - μ)}^{2}

Variance

s^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - \overline{x})}^{2}

Single data set formulae

Population

Sample

μ = E (x) = \frac{1}{N} \sum_{i = 1}^{N} (m_{i} \cdot f_{i})

Arithmetic mean

\bar{x} = \frac{1}{n} \sum_{i = 1}^{n} (m_{i} \cdot f_{i})

Geometric mean

\tilde{x} = {(\prod_{i = 1}^{n} X_{i})}^{\frac{1}{n}}

σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} {(m_{i} - μ)}^{2} \cdot f_{i}

Variance

s^{2} = \frac{1}{n} \sum_{i = 1}^{n} {(m_{i} - \overline{x})}^{2} \cdot f_{i}

C V = \frac{σ}{μ} \cdot 100

Coefficient of variation

C V = \frac{s}{\overline{x}} \cdot 100

Table B3

Basic probability rules

P (A \cap B) = P (A | B) \cdot P (B)

Multiplication rule

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Addition rule

P (A \cap B) = P (A) \cdot P (B)

or

P (A | B) = P (A)

Independence test

Hypergeometric distribution formulae

n C x = (\binom{n}{x}) = \frac{n!}{x! (n - x)!}

Combinatorial equation

P (x) = \frac{(\binom{A}{x}) (\binom{N - A}{n - x})}{(\binom{N}{n})}

Probability equation

E (X) = μ = n p

Mean

σ^{2} = (\frac{N - n}{N - 1}) n p (q)

Variance

Binomial distribution formulae

P (x) = \frac{n!}{x! (n - x)!} p^{x} {(q)}^{n - x}

Probability density function

E (X) = μ = n p

Arithmetic mean

σ^{2} = n p (q)

Variance

Geometric distribution formulae

P (X = x) = {(1 - p)}^{x - 1} (p)

Probability when

x

is the first success.

Probability when

x

is the number of failures before first success

P (X = x) = {(1 - p)}^{x} (p)

μ = \frac{1}{p}

Mean

μ = \frac{1 - p}{p}

σ^{2} = \frac{(1 - p)}{p^{2}}

Variance

σ^{2} = \frac{(1 - p)}{p^{2}}

Poisson distribution formulae

P (x) = \frac{e^{- μ} μ^{x}}{x!}

Probability equation

E (X) = μ

Mean

σ^{2} = μ

Variance

Uniform distribution formulae

f (x) = \frac{1}{b - a}

for

a \leq x \leq b

PDF

E (X) = μ = \frac{a + b}{2}

Mean

σ^{2} = \frac{{(b - a)}^{2}}{12}

Variance

Exponential distribution formulae

P (X \leq x) = 1 - e^{- m x}

Cumulative probability

E (X) = μ = \frac{1}{m}

or

m = \frac{1}{μ}

Mean and decay factor

σ^{2} = \frac{1}{m^{2}} = μ^{2}

Variance

Table B4

The following page of formulae requires the use of the " $Z$ ", " $t$ ", " $χ^{2}$ " or " $F$ " tables.

Z = \frac{x - μ}{σ}

Z-transformation for normal distribution

Z = \frac{x - n p'}{\sqrt{n p' (q')}}

Normal approximation to the binomial

Probability (ignores subscripts)
Hypothesis testing

Confidence intervals
[bracketed symbols equal margin of error]
(subscripts denote locations on respective distribution tables)

Z_{c} = \frac{\bar{x} - μ_{0}}{\frac{σ}{\sqrt{n}}}

Interval for the population mean when sigma is known

\bar{x} \pm [Z_{(α / 2)} \frac{σ}{\sqrt{n}}]

Z_{c} = \frac{\bar{x} - μ_{0}}{\frac{s}{\sqrt{n}}}

Interval for the population mean when sigma is unknown but $n > 30$

\bar{x} \pm [Z_{(α / 2)} \frac{s}{\sqrt{n}}]

t_{c} = \frac{\bar{x} - μ_{0}}{\frac{s}{\sqrt{n}}}

Interval for the population mean when sigma is unknown but $n < 30$

\bar{x} \pm [t_{(n - 1), (α / 2)} \frac{s}{\sqrt{n}}]

Z_{c} = \frac{p' - p_{0}}{\sqrt{\frac{p_{0} q_{0}}{n}}}

Interval for the population proportion

p' \pm [Z_{(α / 2)} \sqrt{\frac{p' q'}{n}}]

t_{c} = \frac{\bar{d} - δ_{0}}{\frac{s_{d}}{\sqrt{n}}}

Interval for difference between two means with matched pairs

\bar{d} \pm [t_{(n - 1), (α / 2)} \frac{s_{d}}{\sqrt{n}}]

where $s_{d}$ is the deviation of the differences

Z_{c} = \frac{(\bar{x_{1}} - \bar{x_{2}}) - δ_{0}}{\sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}}

Interval for difference between two means when sigmas are known

(\bar{x_{1}} - \bar{x_{2}}) \pm [Z_{(α / 2)} \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}]

t_{c} = \frac{({\bar{x}}_{1} - {\bar{x}}_{2}) - δ_{0}}{\sqrt{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}}

Interval for difference between two means with equal variances when sigmas are unknown

({\bar{x}}_{1} - {\bar{x}}_{2}) \pm [t_{d f, (α / 2)} \sqrt{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}]

where

d f = \frac{{(\frac{{(s_{1})}^{2}}{n_{1}} + \frac{{(s_{2})}^{2}}{n_{2}})}^{2}}{(\frac{1}{n_{1} - 1}) (\frac{{(s_{1})}^{2}}{n_{1}}) + (\frac{1}{n_{2} - 1}) (\frac{{(s_{2})}^{2}}{n_{2}})}

Z_{c} = \frac{({p'}_{1} - {p'}_{2}) - δ_{0}}{\sqrt{\frac{{p'}_{1} ({q'}_{1})}{n_{1}} + \frac{{p'}_{2} ({q'}_{2})}{n_{2}}}}

Interval for difference between two population proportions

({p'}_{1} - {p'}_{2}) \pm [Z_{(α / 2)} \sqrt{\frac{{p'}_{1} ({q'}_{1})}{n_{1}} + \frac{{p'}_{2} ({q'}_{2})}{n_{2}}}]

χ_{c}^{2} = \frac{(n - 1) s^{2}}{σ_{0}^{2}}

Tests for GOF, Independence, and Homogeneity

χ_{c}^{2} = Σ \frac{{(O - E)}^{2}}{E}

where O = observed values and E = expected values

F_{c} = \frac{s_{1}^{2}}{s_{2}^{2}}

Where $s_{1}^{2}$ is the sample variance which is the larger of the two sample variances

The next 3 formulae are for determining sample size with confidence intervals.
(note: E represents the margin of error)

n = \frac{Z_{(\frac{a}{2})}^{2} σ^{2}}{E^{2}}

Use when sigma is known

E = \bar{x} - μ

n = \frac{Z_{(\frac{a}{2})}^{2} (0.25)}{E^{2}}

Use when $p'$ is unknown

E = p' - p

n = \frac{Z_{(\frac{a}{2})}^{2} [p' (q')]}{E^{2}}

Use when $p'$ is unknown

E = p' - p

Table B5

Simple linear regression formulae for $y = a + b (x)$

r = \frac{Σ [(x - \bar{x}) (y - \bar{y})]}{\sqrt{Σ {(x - \bar{x})}^{2} * Σ {(y - \bar{y})}^{2}}} = \frac{S_{x y}}{S_{x} S_{y}} = \sqrt{\frac{S S R}{S S T}}

Correlation coefficient

b = \frac{Σ [(x - \bar{x}) (y - \bar{y})]}{Σ {(x - \bar{x})}^{2}} = \frac{S_{x y}}{{S S}_{x}} = r_{y, x} (\frac{s_{y}}{s_{x}})

Coefficient b (slope)

a = \bar{y} - b (\bar{x})

y-intercept

s_{e}^{2} = \frac{Σ {(y_{i} - {\hat{y}}_{i})}^{2}}{n - k} = \frac{Σ_{i = 1}^{n} e_{i}^{2}}{n - k}

Estimate of the error variance

S_{b} = \frac{s_{e}^{2}}{\sqrt{{(x_{i} - \bar{x})}^{2}}} = \frac{s_{e}^{2}}{(n - 1) s_{x}^{2}}

Standard error for coefficient b

t_{c} = \frac{b - β_{0}}{s_{b}}

Hypothesis test for coefficient β

b \pm [t_{n - 2, α / 2} S_{b}]

Interval for coefficient β

\hat{y} \pm [t_{α / 2} * s_{e} (\sqrt{\frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{s_{x}}})]

Interval for expected value of y

\hat{y} \pm [t_{α / 2} * s_{e} (\sqrt{1 + \frac{1}{n} + \frac{{(x_{p} - \bar{x})}^{2}}{s_{x}}})]

Prediction interval for an individual y

ANOVA formulae

S S R = Σ_{i = 1}^{n} {({\hat{y}}_{i} - \bar{y})}^{2}

Sum of squares regression

S S E = Σ_{i = 1}^{n} {({\hat{y}}_{i} - {\bar{y}}_{i})}^{2}

Sum of squares error

S S T = Σ_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}

Sum of squares total

R^{2} = \frac{S S R}{S S T}

Coefficient of determination

Table B6

The following is the breakdown of a one-way ANOVA table for linear regression.

Source of variation

Sum of squares

Degrees of freedom

Mean squares

F-ratio

Regression

S S R

1

or

k - 1

$M S R = \frac{S S R}{d f_{R}}$

F = \frac{M S R}{M S E}

Error

S S E

$n - k$

$M S E = \frac{S S E}{d f_{E}}$

Total

S S T

$n - 1$

Table B7

B | Mathematical Phrases, Symbols, and Formulas

English Phrases Written Mathematically

Symbols and Their Meanings

Formulas